1 Closures of Relations Based on Aaron Bloomfield Modified by Longin Jan Latecki Rosen, Section 8.4

2 Relational closures Three types we will study –Reflexive Easy –Symmetric Easy –Transitive Hard

3 Consider a relation R: –From our MapQuest example in the last slide set –Note that it is not reflexive We want to add edges to make the relation reflexive By adding those edges, we have made a non- reflexive relation R into a reflexive relation This new relation is called the reflexive closure of R Reflexive closure

4 In order to find the reflexive closure of a relation R, we add a loop at each node that does not have one The reflexive closure of R is R U  –Where  = { (a,a) | a  R } Called the “diagonal relation” –With matrices, we set the diagonal to all 1’s

5 Rosen, section 7.4, question 1(a) Let R be a relation on the set { 0, 1, 2, 3 } containing the ordered pairs (0,1), (1,1), (1,2), (2,0), (2,2), and (3,0) What is the reflexive closure of R? We add all pairs of edges (a,a) that do not already exist We add edges: (0,0), (3,3)

6 Consider a relation R: –From our MapQuest example in the last slide set –Note that it is not symmetric We want to add edges to make the relation symmetric By adding those edges, we have made a non- symmetric relation R into a symmetric relation This new relation is called the symmetric closure of R Symmetric closure

7 In order to find the symmetric closure of a relation R, we add an edge from a to b, where there is already an edge from b to a The symmetric closure of R is R U R -1 –If R = { (a,b) | … } –Then R -1 = { (b,a) | (a,b) in R }

8 Rosen, section 8.4, question 1(b) Let R be a relation on the set { 0, 1, 2, 3 } containing the ordered pairs (0,1), (1,1), (1,2), (2,0), (2,2), and (3,0) What is the symmetric closure of R? We add all pairs of edges (a,b) where (b,a) exists –We make all “single” edges into anti-parallel pairs We add edges: (0,2), (0,3) (1,0), (2,1)

9 Paths in directed graphs A path is a sequences of connected edges from vertex a to vertex b No path exists from the noted start location A path that starts and ends at the same vertex is called a circuit or cycle –Must have length ≥ 1 Start (a) End (b) Start (a) End (b)

10 More on paths… The length of a path is the number of edges in the path, not the number of nodes

11 Shortest paths What is really needed in most applications is finding the shortest path between two vertices

12 Transitive closure The transitive closure would contain edges between all nodes reachable by a path of any length

13 Transitive closure Informal definition: If there is a path from a to b, then there should be an edge from a to b in the transitive closure First take of a definition: –In order to find the transitive closure of a relation R, we add an edge from a to c, when there are edges from a to b and b to c But there is a path from 1 to 4 with no edge! R = { (1,2), (2,3), (3,4) } (1,2) & (2,3)  (1,3) (2,3) & (3,4)  (2,4)

14 Transitive closure Informal definition: If there is a path from a to b, then there should be an edge from a to b in the transitive closure Second take of a definition: –In order to find the transitive closure of a relation R, we add an edge from a to c, when there are edges from a to b and b to c –Repeat this step until no new edges are added to the relation We will study different algorithms for determining the transitive closure red means added on the first repeat teal means added on the second repeat

15 6 degrees of separation The idea that everybody in the world is connected by six degrees of separation –Where 1 degree of separation means you know (or have met) somebody else Let R be a relation on the set of all people in the world –(a,b)  R if person a has met person b So six degrees of separation for any two people a and g means: –(a,b), (b,c), (c,d), (d,e), (e,f), (f,g) are all in R Or, (a,g)  R 6

16 Connectivity relation R contains edges between all the nodes reachable via 1 edge R◦R = R 2 contains edges between nodes that are reachable via 2 edges in R R 2 ◦R = R 3 contains edges between nodes that are reachable via 3 edges in R R n = contains edges between nodes that are reachable via n edges in R R* contains edges between nodes that are reachable via any number of edges (i.e. via any path) in R –Rephrased: R* contains all the edges between nodes a and b when is a path of length at least 1 between a and b in R R* is the transitive closure of R –The definition of a transitive closure is that there are edges between any nodes (a,b) that contain a path between them

17 R* is the star closure of relation R, and it is defined as Def.: The transitive closure of a relation R, t(R), Is the smallest transitive relation containing R. Theorem: t(R) = R*.

18 How long are the paths in a transitive closure? Let R be a relation on set A, and let A be a set with n elements –Rephrased: consider a graph G with n nodes and some number of edges Lemma 1: If there is a path (of length at least 1) from a to b in R, then there is a path between a and b of length not exceeding n Proof preparation: –Suppose there is a path from a to b in R –Let the length of that path be m –Let the path be edges (x 0, x 1 ), (x 1, x 2 ), …, (x m-1, x m ) –That’s nodes x 0, x 1, x 2, …, x m-1, x m –If a node exists twice in our path, then it’s not a shortest path As we made no progress in our path between the two occurrences of the repeated node –Thus, each node may exist at most once in the path

19 How long are the paths in a transitive closure? Proof by contradiction: –Assume there are more than n nodes in the path Thus, m > n Let m = n+1 –By the pigeonhole principle, there are n+1 nodes in the path (pigeons) and they have to fit into the n nodes in the graph (pigeonholes) –Thus, there must be at least one pigeonhole that has at least two pigeons –Rephrased: there must be at least one node in the graph that has two occurrences in the nodes of the path Not possible, as the path would not be the shortest path –Thus, it cannot be the case that m > n If there exists a path from a to b, then there is a path from a to b of at most length n

20 Finding the transitive closure Let M R be the zero-one matrix of the relation R on a set with n elements. Then the zero-one matrix of the transitive closure R* is: Nodes reachable with one application of the relation Nodes reachable with two applications of the relation Nodes reachable with n applications of the relation

21 Rosen, section 8.4, example 7 Find the zero-one matrix of the transitive closure of the relation R given by: 

22 Rosen, section 8.4, example 7  12 3

23 Transitive closure algorithm What we did (or rather, could have done): –Compute the next matrix, where 1 ≤ i ≤ n –Do a Boolean join with the previously computed matrix For our example: –Compute –Join that with to yield –Compute –Join that with from above

24 Transitive closure algorithm procedure transitive_closure (M R : zero-one n  n matrix) A := M R B := A for i := 2 to n begin A := A M R B := B  A end { B is the zero-one matrix for R* }  What is the time complexity?O(n4)O(n4)

25 Roy-Warshall Algorithm Uses only O(n 3 ) operations! Procedure Warshall (M R : rank-n 0-1 matrix) W := M R for k := 1 to n for i := 1 to n for j := 1 to n w ij := w ij  (w ik  w kj ) return W {this represents R * } w ij = 1 means there is a path from i to j going only through nodes ≤k. Indices i and j may have index higher than k.

26 Example of Roy-Warshall Algorithm Example 8 on p. 551 in Rosen’s book.